"strip of wood, bar nailed across parallel boards to hold them together," 1650s, Englished variant of baton "a stick, a staff" (see baton). The nautical sense of "strip of wood nailed down over a tarpaulin over a ship's hatches to prevent leakage in stormy weather" is attested from 1769.
1530s, "action of battering," in law, "the unlawful beating of another," from French batterie, from Old French baterie "beating, thrashing, assault" (12c.), from batre "to beat," from Latin battuere (see batter (v.)).
The meaning shifted in French from "bombardment" ("heavy blows" upon city walls or fortresses) to "unit of artillery" (a sense recorded in English from 1550s). The extension to "electrical cell" (1748, in Ben Franklin) is perhaps from the artillery sense via notion of "discharges" of electricity. In Middle English, bateri meant only "forged metal ware." In obsolete baseball jargon battery was the word for "pitcher and catcher" considered as a unit (1867, originally only the pitcher).
"small open vessel (smaller than a ship) used to cross waters, propelled by oars, a sail, or (later) an engine," Old English bat, from Proto-Germanic *bait- (source also of Old Norse batr, Dutch boot, German Boot), possibly from PIE root *bheid- "to split," if the notion is of making a boat by hollowing out a tree trunk or from split planking. Or it may be an extension of the name for some part of a ship.
French bateau "boat" is from Old English or Norse. Spanish batel, Italian battello, Medieval Latin batellus likewise probably are from Germanic languages. Of serving vessels resembling a boat, by 1680s (ship for "serving vessel or utensil shaped like a ship" is attested by 1520s). The image of being in the same boat "subject to similar challenges and difficulties" is by 1580s; to rock the boat "disturb stability" is from 1914.